I've asked a very similar question also at math.stackexchange, but I've not received any answer.
A vectorial function $\boldsymbol{x}:\mathbb{R}^D \rightarrow \mathbb{R}^N$ satisfies the following linear PDE
$$ \sum_{i=1}^D c_i \frac{\partial \boldsymbol{x}(t_1, \dots, t_D)}{\partial t_i} = \boldsymbol{A}(t_1, \dots, t_D) \boldsymbol{x}(t_1, \dots, t_D). $$
The scalars $c_i$ are constant, and $\boldsymbol{A}(t_1, \dots, t_D): \mathbb{R}^D \rightarrow \mathbb{R}^{N \times N}$ is quasiperiodic, i.e.
$$ \boldsymbol{A}(t_1, \dots, t_i+T_i, \dots, t_D) = \boldsymbol{A}(t_1, \dots, t_i, \dots, t_D). $$
Which boundary conditions should I impose in order to numerically find one of its fundamental solution matrices?